solve for limits of 1+ and 1- for this function:
(x^2 + abs (x-1) - 1)/ abs (x-1)
i don't know how to simplify abs. value expression times polynomial. can someone please help.
Help with this one ?
as x->1+, |x-1| = x-1, so
(x^2+|x-1|-1)/|x-1|
= (x^2+x-2)/(x-1)
= (x+2)(x-1)/(x-1)
= x+2
so limit = 1+2 = 3
as x->1-, |x-1| = -(x-1), so
(x^2+|x-1|-1)/|x-1|
= (x^2-x)/-(x-1)
= x(x-1)/-(x-1)
= -x
so limit = -1
To find the limits of a function as x approaches a certain value, we evaluate the function from both the right-hand side (1+) and the left-hand side (1-) of that value.
For the given function, (x^2 + |x-1| - 1)/ |x-1|, we can simplify the expression to make it easier to evaluate the limits.
1. Let's focus on the absolute value term, |x-1|. This is defined as x-1 if x ≥ 1, and -(x-1) if x < 1. We can rewrite the function accordingly:
- If x ≥ 1, then |x-1| = x-1.
- If x < 1, then |x-1| = -(x-1) = 1-x.
2. Now, substitute these values back into the original function:
- If x ≥ 1, the function becomes (x^2 + (x-1) - 1)/(x-1).
- If x < 1, the function becomes (x^2 + (1-x) - 1)/(1-x).
3. We can simplify each case further:
- For x ≥ 1:
- Numerator: x^2 + x - 2.
- Denominator: x - 1.
- For x < 1:
- Numerator: x^2 - x.
- Denominator: 1 - x.
Now, we can evaluate the limits separately for x approaching 1 from the right (1+) and left (1-).
1. Limit as x approaches 1+:
- This means x is getting closer and closer to 1 from values greater than 1.
- Evaluate the function by substituting x = 1 into the expression for x ≥ 1:
- Numerator: 1^2 + 1 - 2 = 0.
- Denominator: 1 - 1 = 0.
- Division by zero is undefined, so the limit does not exist as x approaches 1 from the right.
2. Limit as x approaches 1-:
- This means x is getting closer and closer to 1 from values less than 1.
- Evaluate the function by substituting x = 1 into the expression for x < 1:
- Numerator: 1^2 - 1 = 0.
- Denominator: 1 - 1 = 0.
- Division by zero is undefined, so the limit does not exist as x approaches 1 from the left.
In conclusion, the limits of the given function as x approaches 1 from both the right and left are undefined.